Remark 5.5 
Classical bifurcation theory implies the existence of a locally quadratic, normally hyperbolic and attracting critical manifold \(S^2\)  for the second iterate (layer) map \(v \mapsto \bar{v}^2\) . The manifold \(S^2\) 
intersects \(S\)  transversally at the supercritical flip point \(Q\) , see Figure REF . Dynamically, \(S^2\)  forms a manifold of stable period-2 cycles for the layer map (REF ). In fact there is a period-doubling cascade for increasing \(w\)  which leads to chaotic dynamics in the layer map. We do not consider chaotic dynamics in detail in this work, but refer to [1]} for details on chaotic dynamics in the layer map (REF ) and [2]} for details on the slow passage through a period-doubling cascade.

